The job-search problem with competition: an evolutionarily stable dynamic strategy

We consider a game-theoretical solution for an optimal stopping problem which we describe in terms of a job-search problem with an infinite population of candidates and an infinite population of posts of varying value. Each candidate finds posts from the post population at unit rate. If a post found is still vacant, a candidate can either accept it or reject it. The reward to a candidate is the value of the post if one is accepted and zero if he never accepts a post. There are no costs for searching, no discounting of future rewards and no recall of previously found posts. The only pressure on a candidate to accept a post comes from the changing rate at which he finds vacant posts (and the values associated with them) as a result of the actions of the other candidates. It is not possible to define optimality for a single candidate without reference to the policies followed by the other candidates. We say a policy . is an evolutionarily stable strategy (ESS) if it has the following property: if all candidates used . and an individual candidate was given the option of changing his policy, then it would not be to his advantage to do so. We first find the optimal value function and optimal policy for the case of a single candidate operating in an environment where the distribution of posts on offer and the chance of finding one both vary with time in a known way. We then show that for the infinite population there is a unique ESS given by a control-limit policy . c, where the control-limit function c is the solution of a given differential equation with a given initial condition. This function c also gives the expected future reward function for any single candidate when all candidates use . c.